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CHAPTER 7: DEVELOPING THE LINEAR PROGRAMME AND SOLVING GRAPHICALLY

This chapter discusses the tools required to solve linear programming problems, including linear functions, graphs, simultaneous linear equations, and the MS-Excel Solver software package. It also covers the formulation and solving of a linear programming problem to maximize profit, with a set of constraints and non-negative constraints.

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CHAPTER 7: DEVELOPING THE LINEAR PROGRAMME AND SOLVING GRAPHICALLY

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  1. CHAPTER 7: DEVELOPING THE LINEAR PROGRAMME AND SOLVING GRAPHICALLY

  2. TOOLS REQUIRED TO SOLVE A LINEAR PROGRAMMING • linear functions • graphs and co-ordinate systems • representing linear functions graphically • solving simultaneous linear equations • Software package—MS-Excel Solver

  3. FORMULATING THE LINEAR PROGRAMMING • The definition of the decision variables • Let X = the number of Tables manufactured per week • Let Y = the number of Chairs manufactured per week • The objective function • Maximise Profit = 4X + 3Y

  4. The set of constraints • 4X + 1Y  90 [Constraint due to Wood] • 2X + 1Y  50 [Constraint due to Machine-Time] • 1X + 1Y  40 [Constraint due to Polishing-Time] • X0, Y  0 [non-negative constraint]

  5. SUMMARY • Let X = the number of Tables made per week, Let Y = the number of Chairs made per week, • Maximise Profit = 4X + 3Y Objective Function Subject to 4X+1Y  90 Wood 2X+1Y  50 Machine-Time 1X +1Y  40 Polishing-Time X, Y  0

  6. SOLVING A LINEAR PROGRAMME • Stage 1 • 4X+1Y  90 Wood constraints

  7. 2X+1Y  50 Machine-Time

  8. 1X +1Y  40 Polishing-Time

  9. FEASIBLE REGION (O, A, B. C, D)--the set of all possible solutions to satisfy all the constraints.

  10. X Y Profit 0 0 0 Read off graph 0 40 120 Read off graph 10 30 130 Solving, see i) below. 20 10 110 Solving, see ii) below. 22.5 0 90 Read off graph • Stage 2 • The conclusion is that the optimal solution, the value of X and Y that maximises the profit function, must lie at one of the corner points.

  11. i) This corner point is the intersection of the Machine-Time constraint line and the Polishing-Time constraint line 2X + Y = 50 X + Y = 40 (X, Y) = (10, 30) • ii) This corner point is the intersection of the Wood and the Machine-Time constraints 4X + Y = 90 2X + Y = 50 (X, Y) = (20, 10)

  12. INTERPRETING THE SOLUTION • The Solution to the linear programming can be stated is: The firm should manufacture 10 Tables and 30 Chairs per week to make the maximum possible weekly profit of £130.

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